This talk is an overview of my joint work with Dima Yafaev during 2013-2015, published as a series of five papers. We identify a class of compact Hankel operators for which the singular values have a power asymptotics. More precisely, for every operator in this class, its n'th singular value behaves as a constant times some negative power of n as n goes to infinity. The constant can be explicitly computed. This class of Hankel operators can be described in four different representations: as (infinite) Hankel matrices, as integral Hankel operators on the half-line, and as operators on the Hardy classes on the unit circle and on the real line. Each representation brings to the fore a different aspect of the problem.